Question: The sum of two numbers is $93$, and their difference is $13$. What are the two numbers?
Solution: Let $x$ be the first number, and let $y$ be the second number. The system of equations is: ${x+y = 93}$ ${x-y = 13}$ Solve for $x$ and $y$ using elimination. Add the top and bottom equations together. $ 2x = 106 $ $ x = \dfrac{106}{2} $ ${x = 53}$ Now that you know ${x = 53}$ , plug it back into $ {x+y = 93}$ to find $y$ ${(53)}{ + y = 93}$ ${y = 40}$ You can also plug ${x = 53}$ into $ {x-y = 13}$ and get the same answer for $y$ ${(53)}{ - y = 13}$ ${y = 40}$ Therefore, the larger number is $53$, and the smaller number is $40$.